Problem 1 - Algebra 2 and Trigonometry

Question

Calvin said that the graph of \(y=\tan \left(x-\frac{\pi}{4}\right)\) has asymptotes at \(x=\frac{3 \pi}{4}+n \pi\) for all integral values of \(n .\) Do you agree with Calvin? Explain why or why not.

Step-by-Step Solution

Answer

Yes, Calvin is correct; the asymptotes are at \( x=\frac{3\pi}{4}+n\pi \).

Step 1: Understand the Parent Function

The function we are analyzing is based on the tangent function, which is generally written as \( y = \tan(x) \). The tangent function has vertical asymptotes at \( x = \frac{\pi}{2} + n\pi \), where \( n \) is any integer.

Step 2: Identify the Transformation

The given function is \( y = \tan(x - \frac{\pi}{4}) \). This is a horizontal shift of \( \tan(x) \) to the right by \( \frac{\pi}{4} \) units. This transformation affects the positions of the asymptotes.

Step 3: Calculate New Asymptote Positions

To find the new positions of the asymptotes after the shift, we set \( x - \frac{\pi}{4} = \frac{\pi}{2} + n\pi \) (the original asymptote positions). Solving for \( x \), we add \( \frac{\pi}{4} \) to both sides to get \( x = \frac{\pi}{2} + \frac{\pi}{4} + n\pi = \frac{3\pi}{4} + n\pi \).

Step 4: Conclusion

The calculated positions of the asymptotes for the transformed function are \( x = \frac{3\pi}{4} + n\pi \). This matches Calvin's statement, confirming that he is correct.

Key Concepts

Tangent Function

The tangent function, denoted as \( y = \tan(x) \), is a fundamental trigonometric function that plays an essential role in numerous mathematical applications. Unlike functions such as sine and cosine, which have a sinusoidal wave shape, the tangent function is characterized by its repeating pattern of steep ups and downs.

The graph of \( y = \tan(x) \) has an intriguing feature: vertical asymptotes. These are invisible vertical lines that represent points where the function tends to infinity. For the parent tangent function, these asymptotes occur at multiples of \( \frac{\pi}{2} + n\pi \), with \( n \) being any integer. In simpler terms:

• The function is undefined at these points.
• That's because the tangent of \( x \) is essentially the sine of \( x \) divided by the cosine of \( x \), and division by zero is undefined.

These key characteristics of the tangent function make it important to consider its transformations when analyzing more complex expressions.

Vertical Asymptotes

Vertical asymptotes are a vital concept when working with functions like the tangent function. They denote the points where the function grows without bound, both positively and negatively, and thus are places where the function does not exist.

For the tangent function \( y = \tan(x) \), the vertical asymptotes are located at:

• \( x = \frac{\pi}{2} + n\pi \), where \( n \) is any integer.
• These occur because the cosine of \( x \) is zero at these points, making \( \tan(x) \) undefined.

When transformations such as horizontal shifts are applied to \( \tan(x) \), the positions of these asymptotes change as well.

In the given function \( y = \tan(x - \frac{\pi}{4}) \), the asymptotes shift due to the function's adjusted baseline. Specifically, the asymptotes of the parent function \( \tan(x) \) are moved to \( x = \frac{3\pi}{4} + n\pi \), aligning with Calvin's observation. Understanding how transformations affect vertical asymptotes is crucial for correctly graphing and interpreting functions.

Horizontal Shift

A horizontal shift is a specific type of transformation applied to a function, moving it left or right along the x-axis. For trigonometric functions like the tangent, this shift alters the points where specific features such as asymptotes appear.

In the context of the tangent function, \( y = \tan(x - \frac{\pi}{4}) \) represents a horizontal shift. This transformation can be understood from:

• The parent function, \( \tan(x) \), being shifted to the right by \( \frac{\pi}{4} \) units.
• As a result, any characteristic of the \( \tan(x) \) function, including vertical asymptotes, moves in tandem.

To find the new asymptote positions, one must account for this shift: start from the original asymptote formula \( x = \frac{\pi}{2} + n\pi \), and adjust with the shift amount, adding \( \frac{\pi}{4} \) to each term. The new equation becomes \( x = \frac{3\pi}{4} + n\pi \), confirming how the transformation affects the graph. Understanding horizontal shifts is vital for adjusting graphs correctly in function transformations.